To save energy and alleviate interferences in a wireless sensor network, the usage of virtual backbone was proposed. Because of accidental damages or energy depletion, it is desirable to construct a fault tolerant virtual backbone, which can be modeled as a k-connected m-fold dominating set (abbreviated as (is adjacent with at least m nodes in C and the subgraph of G induced by C is k-connected. In this paper, we present an approximation algorithm for the minimum (3, m)-CDS problem with m ≥ 3. The performance ratio is at most γ, where γ = α + 8 + 2 ln(2α − 6) for α ≥ 4 and γ = 3α + 2 ln 2 for α < 4, and α is the performance ratio for the minimum (2, m)-CDS problem. Using currently best known value of α, the performance ratio is ln δ + o(ln δ), where δ is the maximum degree of the graph, which is asymptotically best possible in view of the non-approximability of the problem. This is the first performance-guaranteed algorithm for the minimum (3, m)-CDS problem on a general graph. Furthermore, applying our algorithm on a unit disk graph which models a homogeneous wireless sensor network, the performance ratio is less than 27, improving previous ratio 62.3 by a large amount for the (3, m)-CDS problem on a unit disk graph.
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